The imitation of the three-dimensional arts of plaiting, weaving
and basketry was the origin of interlaced and knotwork
interlaced designs. There are few races who have not used it as a
decoration of stone, wood and metal. Interlacing rosettes,
friezes and ornaments are to be found in the art of most peoples
surrounding the Mediterranean, the Black and Caspian Seas,
Egyptians, Greeks, Romans, Byzantines, Moors, Persians, Turks,
Arabs, Syrians, Hebrews and African tribes. Their highlights are
Celtic interlacing knotworks, Islamic layered patterns and
Moorish floor and wall decorations.

Their common geometrical construction principle, discovered by
P.Gerdes, is the use of (two-sided) mirrors incident to
the edges of a square, triangular or hexagonal regular plane
tiling, or perpendicular to its edges in their
midpoints. In the ideal case, after the series of consecutive
reflections, the ray of light reaches its beginning point,
defining a single closed curve. In other cases, the result
consists of several such curves.

For example, to the Celtic designs from G.Bain book "Celtic
Art" correspond the following mirror-schemes:

Trying to discover their common mathematical background, they
appear two questions: how to construct such a
perfect curve - a single line placed uniformly in a regular
tiling, this means, how to arrange the set of mirrors generating
it, and how to classify the curves obtained. In
principle, any polyomino (polyiamond or polyhexe) with mirrors
on its border, and two-sided mirrors between cells
or perpendicular to the internal cell-edges in their
midpoints, could be used for the creation of the
corresponding curves.

For their construction in some polyomino (polyiamond or polyhexe),
we propose the following method. First, we construct
all the different curves in it without using internal
mirrors, starting from different
cell-edge midpoints and ending in them, till the polyomino is
exhausted, i.e. uniformly covered by k curves. After that,
we may use "curve surgery" in order to obtain a single curve,
according to the following rules:

any mirror introduced in a crossing point of two distinct
curves connects them into one curve;

depending on the position of a mirror, a mirror introduced
into a self-crossing point of an (oriented) curve not changes the
number of curves, or breaks the curve into two closed
curves.

In every polyomino we may introduce k-1, k,
k+1,..., 2A-P/2 internal two-sided
mirrors, where A is the area and P is the
perimeter of the polyomino. Introducing the minimal number
of mirrors k-1, we first obtain a single curve,
and in the next steps we try to preserve that result.

In the case of a rectangular square grid RG[a,b]
with the sides a, b, the initial number of
curves, obtained without using internal mirrors
is k = gcd(a,b) (greatest
common divisor), so in order
to obtain a single curve, the possible
number of internal two-sided mirrors is k-1, k,...,
2ab-a-b. According to the rules
for introduction of internal mirrors, we
propose the following algorithm for the production of
monolinear designs: in every step, each from the
first internal k-1 mirrors must be introduced in crossing
points belonging to different curves. After that, when the curves
are connected and transformed into a single line, we may
introduce other mirrors, taking care about the number of curves,
according to the rules mentioned. For example:

The symmetry of such curves is used for the classification of the
Celtic frieze designs by P.Cromwel, and for the reconstruction
of Tamil designs by P.Gerdes.

From the ornamental heritage, at first glance it
looks that the symmetry is the mathematical
basis for their construction and possible classification. But,
the existence of such asymmetrical curves suggests the
other approach.

First criterion that we may use is the geometrical one: two curves
are equal iff there is a similarity transforming one into the
other. This means, that one curve can be obtained from the other
by a combined action of a proportionality and isometry.
Instead of considering the curves, we may consider the equal
mirror arrangements defined in the same way. Having the algorithm
for the construction of such perfect curves and the criterion for
their equality, we may try to enumerate them: to find the number
of all the different curves (i.e. mirror arrangements) which can
be derived from a rectangle with the sides a, b,
for a given number of mirrors m
(m = k-1, k, ..., 2ab-a-b).

The other point of view to the classification of such perfect
curves is that of the knot theory. Every such curve can be
simply transformed into an interlacing knotwork design, this
means, into the projection of some alternating knot.